Solve for y
y=3
y=-7
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y^{2}+4y+4=25
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+2\right)^{2}.
y^{2}+4y+4-25=0
Subtract 25 from both sides.
y^{2}+4y-21=0
Subtract 25 from 4 to get -21.
a+b=4 ab=-21
To solve the equation, factor y^{2}+4y-21 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.
-1,21 -3,7
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.
-1+21=20 -3+7=4
Calculate the sum for each pair.
a=-3 b=7
The solution is the pair that gives sum 4.
\left(y-3\right)\left(y+7\right)
Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.
y=3 y=-7
To find equation solutions, solve y-3=0 and y+7=0.
y^{2}+4y+4=25
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+2\right)^{2}.
y^{2}+4y+4-25=0
Subtract 25 from both sides.
y^{2}+4y-21=0
Subtract 25 from 4 to get -21.
a+b=4 ab=1\left(-21\right)=-21
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-21. To find a and b, set up a system to be solved.
-1,21 -3,7
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.
-1+21=20 -3+7=4
Calculate the sum for each pair.
a=-3 b=7
The solution is the pair that gives sum 4.
\left(y^{2}-3y\right)+\left(7y-21\right)
Rewrite y^{2}+4y-21 as \left(y^{2}-3y\right)+\left(7y-21\right).
y\left(y-3\right)+7\left(y-3\right)
Factor out y in the first and 7 in the second group.
\left(y-3\right)\left(y+7\right)
Factor out common term y-3 by using distributive property.
y=3 y=-7
To find equation solutions, solve y-3=0 and y+7=0.
y^{2}+4y+4=25
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+2\right)^{2}.
y^{2}+4y+4-25=0
Subtract 25 from both sides.
y^{2}+4y-21=0
Subtract 25 from 4 to get -21.
y=\frac{-4±\sqrt{4^{2}-4\left(-21\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-4±\sqrt{16-4\left(-21\right)}}{2}
Square 4.
y=\frac{-4±\sqrt{16+84}}{2}
Multiply -4 times -21.
y=\frac{-4±\sqrt{100}}{2}
Add 16 to 84.
y=\frac{-4±10}{2}
Take the square root of 100.
y=\frac{6}{2}
Now solve the equation y=\frac{-4±10}{2} when ± is plus. Add -4 to 10.
y=3
Divide 6 by 2.
y=-\frac{14}{2}
Now solve the equation y=\frac{-4±10}{2} when ± is minus. Subtract 10 from -4.
y=-7
Divide -14 by 2.
y=3 y=-7
The equation is now solved.
\sqrt{\left(y+2\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
y+2=5 y+2=-5
Simplify.
y=3 y=-7
Subtract 2 from both sides of the equation.
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Simultaneous equation
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Differentiation
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Integration
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Limits
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